Complex-weight separation of Harish-Chandra characters

Suppose $q=\kappa -w\kappa \in Q$. Since $q$ is real, $b=wb$, so $w\in W_b$. Move $b$ into the closed positive chamber and conjugate $a,b,\kappa ,w,q$ by the same Weyl element; lattice membership is preserved. With $I$ and $E_I$ as in [F2], $w\in W_I$, and each generator of $W_I$ fixes $E_I^{\perp }$. Therefore $a-wa\in E_I$; since $q=a-wa\in Q$, [F2] gives $q\in Q_I=Q\cap E_I$. The affine transformation $t_{q}w$ of the affine Weyl group $Q_I⋊W_I$ of the dual subsystem ${\mathrm{\Phi }}_I^{\vee }$ fixes the orthogonal projection $a_I$ of $a$ to $E_I$.

In any locally finite affine root-wall arrangement, the stabilizer of a point $a_I$ is generated by the reflections in walls through that point. To see this, choose a ball about $a_I$ meeting no wall not containing $a_I$. Inside it, the finitely many hyperplanes through $a_I$ cut local chambers; generic line segments join any two local chambers by crossing one wall at a time and avoiding their codimension-two intersections. The corresponding wall reflections therefore act transitively on the local chambers. If an affine Weyl element fixes $a_I$, choose a local chamber $C$ and a product $h$ of these reflections carrying $C$ to its image. Then $h^{-1}g$ preserves the global alcove containing $C$, so it is the identity by the simple transitivity in [F3]. Hence $g=h$. Apply this to $g=t_{q}w$ from [step 1.1] in the arrangement for the dual subsystem ${\mathrm{\Phi }}_I^{\vee }$. Every generating wall has a root $\alpha \in {\mathrm{\Phi }}_I$ and $⟨a,{\alpha }^{\vee }⟩\in \mathbb{Z}$; because $\alpha$ is in ${\mathrm{\Phi }}_I$, [F2] gives $⟨b,{\alpha }^{\vee }⟩=0$, so $⟨\kappa ,{\alpha }^{\vee }⟩\in \mathbb{Z}$. Taking finite parts shows that $w$ belongs to the subgroup $W_{\mathrm{i}\mathrm{n}\mathrm{t}}$ generated by the reflections $s_{\alpha }$ for roots with integral $\kappa$-pairing. Undo now the temporary Weyl conjugation of step 1.1: if that conjugating element was $u$, a generating root $\alpha$ in the conjugated coordinates returns to $u^{-1}\alpha$, and $⟨u\kappa ,{\alpha }^{\vee }⟩=⟨\kappa ,(u^{-1}\alpha {)}^{\vee }⟩$. Thus the original $w$ lies in the integral-root reflection subgroup for the original $\kappa$. From this point on $\kappa ,w,q,{\mathrm{\Phi }}^{+}$ and $Q_{+}$ refer to the original data. No preservation of antidominance or of the positive cone by $u$ is assumed.

The set ${\mathrm{\Phi }}_{\mathrm{i}\mathrm{n}\mathrm{t}}=\{\alpha \in \mathrm{\Phi }:⟨\kappa ,{\alpha }^{\vee }⟩\in \mathbb{Z}\}$ is stable under its root reflections: if $m=⟨\kappa ,{\alpha }^{\vee }⟩\in \mathbb{Z}$ and $\beta \in {\mathrm{\Phi }}_{\mathrm{i}\mathrm{n}\mathrm{t}}$, then $⟨\kappa ,(s_{\alpha }\beta {)}^{\vee }⟩=⟨s_{\alpha }\kappa ,{\beta }^{\vee }⟩=⟨\kappa ,{\beta }^{\vee }⟩-m⟨\alpha ,{\beta }^{\vee }⟩\in \mathbb{Z},$ using crystallographic integrality. It is therefore a finite root system in its span, with positive system ${\mathrm{\Phi }}_{\mathrm{i}\mathrm{n}\mathrm{t}}\cap {\mathrm{\Phi }}^{+}$ and its own simple roots ${\mathrm{\Delta }}_{\mathrm{i}\mathrm{n}\mathrm{t}}$. For every $\delta \in {\mathrm{\Delta }}_{\mathrm{i}\mathrm{n}\mathrm{t}}$, antidominance makes the integer $⟨\kappa ,{\delta }^{\vee }⟩$ nonpositive. If $w=s_{{\delta }_1}\cdots s_{{\delta }_m}$ is reduced in $W_{\mathrm{i}\mathrm{n}\mathrm{t}}$, telescoping gives $\kappa -w\kappa ={\sum }_{j=1}^m⟨\kappa ,{\delta }_j^{\vee }⟩s_{{\delta }_1}\cdots s_{{\delta }_{j-1}}{\delta }_j\mathrm{.}$ Every prefix root is positive in ${\mathrm{\Phi }}_{\mathrm{i}\mathrm{n}\mathrm{t}}$ by [F4], hence is positive in the ambient system; every term lies in $-Q_{+}$. Thus $q\in -Q_{+}$. If $\kappa$ is regular, every integral pairing here is nonzero, so every coefficient is strictly negative.

Combining [step 1.1] and [step 3.1], an assumed $q\in Q_{+}\setminus \{0\}$ also lies in $-Q_{+}$, contradicting [F1]. This proves the antidominant assertion. If $\kappa$ is regular, every integral root pairing in [step 3.1] is strictly negative. For any nonidentity $w\in W_{\mathrm{i}\mathrm{n}\mathrm{t}}$, its reduced word has at least one term, and each term is a strictly negative integer multiple of a positive root; the sum is nonzero because $Q_{+}$ is pointed. Hence $q=0$ forces $w=1$, proving the regular assertion.